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A297672
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Array with four columns read by rows: T(n,k) = number of n step walks in the first octant on a square lattice with last step being right (k=1), left (k=2), up (k=3) or down (k=4).
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0
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1, 0, 0, 0, 1, 1, 1, 0, 3, 1, 1, 1, 6, 6, 6, 2, 20, 10, 10, 10, 50, 50, 50, 25, 175, 105, 105, 105, 490, 490, 490, 294, 1764, 1176, 1176, 1176, 5292, 5292, 5292, 3528, 19404, 13860, 13860
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OFFSET
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1,9
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COMMENTS
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For n >= 1, the ratio of the numbers of right or up last steps to left or down last steps is floor((n+2)/2): floor(n/2). - Roger Ford, Oct 28 2019
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LINKS
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FORMULA
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n=1: T(1,1)=1, T(1,2)=0, T(1,3)=0, T(1,4)=0;
n>1: T(n,1) = C(n,floor(n/2))*C(n-1,floor((n-1)/2)) - C(n,floor((n-1)/2))*C(n-1,floor((n-2)/2));
T(n,2) = T(n,3) = C(n-1,floor(n/2)-1)*C(n,floor(n/2)-1)/floor(n/2);
n odd: T(n,4) = T(n,2);
n even: T(n,4) = T(n,2)*((n/2-1)/(n/2+1));
For n > 1, T(n,2) = T(n,3) = A001263(n,floor(n/2)).
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EXAMPLE
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k= 1 2 3 4 total
N right left up down walks
1 1 0 0 0 =1
2 1 1 1 0 =3
3 3 1 1 1 =6
4 6 6 6 2 =20
There are 6 walks of 4 steps in the octant with the last step right. T(4,1)=6 RRRR, RRLR, RLRR, RUDR, RURR, RRUR.
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CROSSREFS
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KEYWORD
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nonn,tabf,walk,more
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AUTHOR
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STATUS
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approved
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